Demonstration of certain Formulce. 117 



been tried, and we thence infer that it will hold for all that 

 can be tried. 



Again, Mr. Herapath finds that with a little contrivance, 

 without the admission of principles essentially new, he can 

 show its application to (a:+j/)''' " when r -{- v = n, and r, r» 

 and n are whole numbers. So far, viewed as a series of sim- 

 ple inductions, and resting on the evidence of numerous trials, 

 the results are satisfactory. 



Here, however, Mr. Herapath is obliged to pause, and to 

 call in the aid of a new principle on which to found his in- 

 ferences; viz. — that because it was true when r and v were in- 

 tegers, it must be true when they are fractions, and even when 

 they are irrationals and imaginaries too ! — the only condition 

 to which these fractions, irrationals, and imaginaries are sub- 

 jected being, that their aggregate shall be an integer. From 

 this again, he discovers that because it is true of two such 

 quantities, it must also be true of one ; and hence that it is 

 true universally. 



Had the former steps been demonstrated in the usually 

 strict meaning of the tei-m, still it must be admitted that this 

 latter step is altogether gratuitous and unwarranted ; perfectly 

 outre to the spirit of that pure logic of which the mathematical 

 sciences are esteemed the very essence. The investigation 

 lays it down as an essential datum that r and v shall be inte- 

 gers : and immediately proceeds upon the supposition of their 

 being fractions, irrationals, or imaginaries, taken at pleasure ! 

 I admire the mind that can readily generalize particular re- 

 sults ; it is always a mind of great capability : but it is asking 

 from us too much to require our assent to a proposition which 

 rests upon the authority of a demonstration like this. Why 

 not at once assume n as the representative of any number 

 whatever, even an imaginary one (if that does not involve some 

 absurdity), rather than adopt the tedious process of considering 

 ;• and v as both fractions? It was thus that Newton proceededj 

 and he verified the formula by numerous trials. Others since 

 his time have involved the theorem in a maze of symbols, from 

 which they have dexterously evolved a medley of conclusions, 

 which they call demonstrations : but, so far as the proof of the 

 theorem is concerned, the matter rests just where he left it, 

 save that it has received numberless numerical verifications in 

 its passage through the hands of successive computists. 



The " demofistratio?i" given by Mr. Herapath, of 



A"^-'' = {x + nf - n{x + n\f +..., p. 326, 

 also fails, in consequence of his failure in the demonstration of 

 the binomial theorem: and against thai gent Icmaii's reason- 

 ins 



